Solution:- Combining Rational numbers and irrational numbers:

Rational Numbers:- Numbers which either terminating or none-terminating Repeating. and can be convertible in the form of \(\displaystyle\frac{{p}}{{q}}\) where \(\displaystyle{p},{q}\in{R}\) and \(\displaystyle{q}\ne{0}\) is known as Rational numbers

Irrational numbers:- The Numbers which are non- terminating and non-repeating and all square root of prime numbers is known as Irrational numbers.

Yes \(\displaystyle{12}+\sqrt{{2}}\) is irrational number.

Yes \(\displaystyle{12}\cdot\sqrt{{2}}=\frac{{1}}{\sqrt{{2}}}\) is irrational number.

a) prove that sum of rational number and irratoional number is irrational.

Given that :− r be a rational number and t be an irrational number.To prove:− sum of rational number and irratoional number is irrational.

i.e. (r+t)=irrational

Proof:−It is given that r is rational, t is irrational,and assume that is (r+t) rational.

Since a and a+b are rational, we can write them as fraction. Let, r=ap and (r+t)=bp'

\(\displaystyle\Rightarrow{a}{p}+{t}={b}{p}'\)

\(\displaystyle\Rightarrow{t}={b}{p}'−{a}{p}\)

\(\displaystyle\Rightarrow{t}={b}{p}'+{\left(−{a}{p}\right)}\)

\(\displaystyle\Rightarrow{t}=\) (rational number)+(rational number)

\(\displaystyle\Rightarrow{t}=\) rational number(sum of two rational numbers is always rational number)

but \(\displaystyle{t}=\) irratational which is contradiction.

Hence (r+t)= Irrational number.

Hence proved.

b) prove that product of rational number and irratoional number is irrational.

Given that :− r be a rational number and t be an irrational number.

To prove:−product of rational number and irratoional number is irrational.

i.e. (r*t)=irrational

Proof:−It is given that r is rational, t is irrational, and assume that is (r*t) rational.

Since a and a*b are rational, we can write them as fraction

r=ap

and PSK(r×t)=bp'

\(\displaystyle\Rightarrow{a}{p}\cdot{t}={b}{p}'\)

\(\displaystyle\Rightarrow{t}=\frac{{{b}{p}'}}{{{a}{p}}}\)

\(\displaystyle\Rightarrow{t}=\frac{{{b}{p}'}}{{{a}{p}}}\)

\(\displaystyle\Rightarrow{t}=\) (rational number)/(rational number)

\(\displaystyle\Rightarrow{t}=\) rational number(rational number of two rational numbers is always rational number)

but t= irratational

which is contradiction.

Hence (r*t)= Irrational number. Hence proved.

Rational Numbers:- Numbers which either terminating or none-terminating Repeating. and can be convertible in the form of \(\displaystyle\frac{{p}}{{q}}\) where \(\displaystyle{p},{q}\in{R}\) and \(\displaystyle{q}\ne{0}\) is known as Rational numbers

Irrational numbers:- The Numbers which are non- terminating and non-repeating and all square root of prime numbers is known as Irrational numbers.

Yes \(\displaystyle{12}+\sqrt{{2}}\) is irrational number.

Yes \(\displaystyle{12}\cdot\sqrt{{2}}=\frac{{1}}{\sqrt{{2}}}\) is irrational number.

a) prove that sum of rational number and irratoional number is irrational.

Given that :− r be a rational number and t be an irrational number.To prove:− sum of rational number and irratoional number is irrational.

i.e. (r+t)=irrational

Proof:−It is given that r is rational, t is irrational,and assume that is (r+t) rational.

Since a and a+b are rational, we can write them as fraction. Let, r=ap and (r+t)=bp'

\(\displaystyle\Rightarrow{a}{p}+{t}={b}{p}'\)

\(\displaystyle\Rightarrow{t}={b}{p}'−{a}{p}\)

\(\displaystyle\Rightarrow{t}={b}{p}'+{\left(−{a}{p}\right)}\)

\(\displaystyle\Rightarrow{t}=\) (rational number)+(rational number)

\(\displaystyle\Rightarrow{t}=\) rational number(sum of two rational numbers is always rational number)

but \(\displaystyle{t}=\) irratational which is contradiction.

Hence (r+t)= Irrational number.

Hence proved.

b) prove that product of rational number and irratoional number is irrational.

Given that :− r be a rational number and t be an irrational number.

To prove:−product of rational number and irratoional number is irrational.

i.e. (r*t)=irrational

Proof:−It is given that r is rational, t is irrational, and assume that is (r*t) rational.

Since a and a*b are rational, we can write them as fraction

r=ap

and PSK(r×t)=bp'

\(\displaystyle\Rightarrow{a}{p}\cdot{t}={b}{p}'\)

\(\displaystyle\Rightarrow{t}=\frac{{{b}{p}'}}{{{a}{p}}}\)

\(\displaystyle\Rightarrow{t}=\frac{{{b}{p}'}}{{{a}{p}}}\)

\(\displaystyle\Rightarrow{t}=\) (rational number)/(rational number)

\(\displaystyle\Rightarrow{t}=\) rational number(rational number of two rational numbers is always rational number)

but t= irratational

which is contradiction.

Hence (r*t)= Irrational number. Hence proved.